50 Easy Algebra Step-by-Step
Negative Integer Exponents
If x is a nonzero real number and n is a natural number, then x
x
n
n
− =^1.
A negative integer exponent on a non-
zero number tells you to obtain the recip-
rocal of the corresponding exponential
expression that has a positive exponent.
Problem Evaluate.
a. 2 −^5
b. ()−^5
c. (..^6 )−^2
d.
3
4
3
⎛
⎝⎜
⎛⎛
⎝⎝
⎞
⎠⎟
⎞⎞
⎠⎠
−
Solution
a. 2 −^5
Step 1. Write the reciprocal of the corresponding positive exponent version
of 2 −^5.
2
1
2
5
5
− =
Step 2. Evaluate 25.
2
1
2
1
22222
1
32
5
5
− ==
222222
=
b. ()−^5
Step 1. Write the reciprocal of the corresponding positive exponent version
of ()− −^5.
()− =
()−
− 5 1
5
Step 2. Evaluate ()−^5.
()− =
()−
=
−
=
−
− =−
1 1
22222 ⋅−⋅− ⋅−⋅−
1
32
1
32
5
5
P
x
x
−−nn≠−^1 n;.xxxnn≠ xxn A negative
exponent does not make a power negative.
As you can see, the negative
exponent did not make the answer
negative; 2 1
32
− (^5) ≠−.
When you evaluate ()−^5 , the answer is negative
because ()^5 is negative. The negative exponent
is not the reason ()−^5 is negative.